Final answer:
Based on the given information, triangle ABC must be an isosceles triangle because it has two pairs of equal angles and at least two sides of equal length.
Step-by-step explanation:
The question is asking to determine the type of triangle ABC based on the given conditions: segment BD bisects angle ABC, and segment AD is congruent to segment CD. Since angle ABC is bisected by segment BD, we know that angles ABD and CBD are equal. Additionally, since AD is congruent to CD, sides AD and CD are of equal length, which means triangle ADC is an isosceles triangle. By the properties of isosceles triangles, angles DAC and DCA are also equal.
With two pairs of equal angles (ABD and CBD, DAC and DCA), triangle ABC must also have two sides of equal length because the angles opposite these sides will be equal. Therefore, triangle ABC has at least two sides of equal length, making it an isosceles triangle.